*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
goal(xs,ys) -> mul0(xs,ys)
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Weak DP Rules:
Weak TRS Rules:
Signature:
{add0/2,goal/2,mul0/2} / {Cons/2,Nil/0,S/0}
Obligation:
Innermost
basic terms: {add0,goal,mul0}/{Cons,Nil,S}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
goal(xs,ys) -> mul0(xs,ys)
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2,5}
by application of
Pre({2,5}) = {1,3,4}.
Here rules are labelled as follows:
1: add0#(Cons(x,xs),y) ->
c_1(add0#(xs,Cons(S(),y)))
2: add0#(Nil(),y) -> c_2()
3: goal#(xs,ys) -> c_3(mul0#(xs
,ys))
4: mul0#(Cons(x,xs),y) ->
c_4(add0#(mul0(xs,y),y)
,mul0#(xs,y))
5: mul0#(Nil(),y) -> c_5()
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
add0#(Nil(),y) -> c_2()
mul0#(Nil(),y) -> c_5()
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Nil(),y) -> c_2():4
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys))
-->_1 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
-->_1 mul0#(Nil(),y) -> c_5():5
3:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Nil(),y) -> c_5():5
-->_1 add0#(Nil(),y) -> c_2():4
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
4:W:add0#(Nil(),y) -> c_2()
5:W:mul0#(Nil(),y) -> c_5()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: mul0#(Nil(),y) -> c_5()
4: add0#(Nil(),y) -> c_2()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys))
-->_1 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
3:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(2,goal#(xs,ys) -> c_3(mul0#(xs,ys)))]
*** 1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Problem (S)
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
We decompose the input problem according to the dependency graph into the upper component
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
and a lower component
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Further, following extension rules are added to the lower component.
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: mul0#(Cons(x,xs),y) ->
c_4(add0#(mul0(xs,y),y)
,mul0#(xs,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_4) = {1,2}
Following symbols are considered usable:
{add0#,goal#,mul0#}
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [4]
p(Nil) = [0]
p(S) = [11]
p(add0) = [2] x2 + [0]
p(goal) = [4] x1 + [2] x2 + [1]
p(mul0) = [0]
p(add0#) = [0]
p(goal#) = [2] x1 + [1] x2 + [1]
p(mul0#) = [1] x1 + [4] x2 + [0]
p(c_1) = [2] x1 + [2]
p(c_2) = [1]
p(c_3) = [1] x1 + [2]
p(c_4) = [2] x1 + [1] x2 + [0]
p(c_5) = [1]
Following rules are strictly oriented:
mul0#(Cons(x,xs),y) = [1] x + [1] xs + [4] y + [4]
> [1] xs + [4] y + [0]
= c_4(add0#(mul0(xs,y),y)
,mul0#(xs,y))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mul0#(Cons(x,xs),y) ->
c_4(add0#(mul0(xs,y),y)
,mul0#(xs,y))
*** 1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: add0#(Cons(x,xs),y) ->
c_1(add0#(xs,Cons(S(),y)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_1) = {1}
Following symbols are considered usable:
{add0,mul0,add0#,goal#,mul0#}
TcT has computed the following interpretation:
p(Cons) = 1 + x1 + x2
p(Nil) = 0
p(S) = 0
p(add0) = 3 + x1 + x2
p(goal) = 1 + 2*x1 + 2*x1^2 + x2
p(mul0) = x1*x2 + 3*x1^2
p(add0#) = 3 + x1
p(goal#) = x1 + 4*x1*x2 + x1^2 + 2*x2 + x2^2
p(mul0#) = x1 + 4*x1*x2 + 4*x1^2
p(c_1) = x1
p(c_2) = 1
p(c_3) = 0
p(c_4) = 0
p(c_5) = 1
Following rules are strictly oriented:
add0#(Cons(x,xs),y) = 4 + x + xs
> 3 + xs
= c_1(add0#(xs,Cons(S(),y)))
Following rules are (at-least) weakly oriented:
mul0#(Cons(x,xs),y) = 5 + 9*x + 8*x*xs + 4*x*y + 4*x^2 + 9*xs + 4*xs*y + 4*xs^2 + 4*y
>= 3 + xs*y + 3*xs^2
= add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) = 5 + 9*x + 8*x*xs + 4*x*y + 4*x^2 + 9*xs + 4*xs*y + 4*xs^2 + 4*y
>= xs + 4*xs*y + 4*xs^2
= mul0#(xs,y)
add0(Cons(x,xs),y) = 4 + x + xs + y
>= 4 + xs + y
= add0(xs,Cons(S(),y))
add0(Nil(),y) = 3 + y
>= y
= y
mul0(Cons(x,xs),y) = 3 + 6*x + 6*x*xs + x*y + 3*x^2 + 6*xs + xs*y + 3*xs^2 + y
>= 3 + xs*y + 3*xs^2 + y
= add0(mul0(xs,y),y)
mul0(Nil(),y) = 0
>= 0
= Nil()
*** 1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
2:W:mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
3:W:mul0#(Cons(x,xs),y) -> mul0#(xs,y)
-->_1 mul0#(Cons(x,xs),y) -> mul0#(xs,y):3
-->_1 mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: mul0#(Cons(x,xs),y) -> mul0#(xs
,y)
2: mul0#(Cons(x,xs),y) ->
add0#(mul0(xs,y),y)
1: add0#(Cons(x,xs),y) ->
c_1(add0#(xs,Cons(S(),y)))
*** 1.1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):2
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1
2:W:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: add0#(Cons(x,xs),y) ->
c_1(add0#(xs,Cons(S(),y)))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: mul0#(Cons(x,xs),y) ->
c_4(mul0#(xs,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_4) = {1}
Following symbols are considered usable:
{add0#,goal#,mul0#}
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [8]
p(Nil) = [1]
p(S) = [1]
p(add0) = [1] x1 + [0]
p(goal) = [8] x1 + [1] x2 + [1]
p(mul0) = [0]
p(add0#) = [1]
p(goal#) = [1] x1 + [1] x2 + [1]
p(mul0#) = [2] x1 + [2] x2 + [0]
p(c_1) = [2] x1 + [1]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [1] x1 + [15]
p(c_5) = [2]
Following rules are strictly oriented:
mul0#(Cons(x,xs),y) = [2] xs + [2] y + [16]
> [2] xs + [2] y + [15]
= c_4(mul0#(xs,y))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
Weak TRS Rules:
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
Weak TRS Rules:
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
-->_1 mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mul0#(Cons(x,xs),y) ->
c_4(mul0#(xs,y))
*** 1.1.1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
Obligation:
Innermost
basic terms: {add0#,goal#,mul0#}/{Cons,Nil,S}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).